5. It is required to convert/2, or its equal✔(1+1), into an infinite series. Ans. 1+ 1 1 3 3.5 &c. 6. It is required to convert 3/7, or its equal (8—1)3, 7. It is required to convert/240, or its equal (243 – 3)3, into an infinite series. 8. It is required to convert (a±x) into an infinite 9. It is required to convert (a±b) into an infinite series. 10. It is required to convert (a-b) into an infinite 11. It is required to convert (a+x) into an infinite series. Ans.a {1+ ·+· 2x x2 4x3 4.7x3 4.7.10x5 → &c.} 12. It is required to convert (1-x) into an infinite 16. It is required to convert (a+x) (a2 — x2)**, into an infinite series. X x2 x3 3x4 3x5 Ans. + -+ ·+ +. 2a2 8a5 150 OF THE INDETERMINATE ANALYSIS. In the common rules of Algebra, such questions are usually proposed as require some certain or definite answer; in which case, it is necessary that there should be as many independent equations, expressing their conditions, as there are unknown quantities to be determined; or otherwise the problem would not be limited. But in other branches of the science, questions frequently arise that involve a greater number of unknown quantities than there are equations to express them; in which instances they are called indeterminate or unlimited problems; being such as usually admit of an indefinite number of solutions; although, when the question is proposed in integers, and the answers are required only in whole positive numbers, they are, in some cases, confined within certain limits, and in others, the problem may become impossible. PROBLEM 1. To find the integral values of the unknown quantities z and y in the equation. ax-by=±c, or ax+by=c. Where a and b are supposed to be given whole numbers, which admit of no common divisor, except when it is also a divisor of c. RULE. 1. Let wh denote a whole, or integral number; and reduce the equation to the form by±c α c-by —wh, or x= wh. α 2. Throw all whole numbers out of that of these two expressions, to which the question belongs, so that the numbers d and e in the remaining parts, may be each less than as then 3. Take such a multiple of one of these last formulæ, corresponding with that above mentioned, as will make the coeficient of y nearly equal to a, and throw the whole numbers out of it as before. Or find the sum or difference of, and the expression α above used, or any multiple of it that comes near ay dy, and the result, in either of these cases, will still be wh, a whole number. 4. Proceed in the same manner with this last result ; and so on, till the coefficient of y becomes remainder some number r; then 1, and the Where p may be o, or any integral number whatever, that makes y positive; and as the value of y is now known, that of x may be found from the given equation, when the question is possible (m). NOTE. Any indeterminate equation of the form ax-by=±c, in which a and b are prime to each other, is always possible, and will admit of an infinite number of answers in whole numbers. (m) This rule is founded on the obvious principle, that the sum difference, or product of any two whole numbers, is a whole number; and that, if a number divides the whole of any other number and a part of it, it will also diyide the remaining part. |